nLab Archimedean ordered integral domain

Contents

Contents

Idea

An Archimedean ordered integral domain is an ordered integral domain that satisfies the Archimedean property.

Properties

Every Archimedean ordered integral domain extension of the integers [x]\mathbb{Z}[x] is a dense linear order.

Since [x]\mathbb{Z}[x] is Archimedean, it does not have either infinite or infinitesimal elements, which means there exists an integer aa and natural number bb such that 0<xa<10 \lt x - a \lt 1 and 1<b(xa)1 \lt b(x - a). 0<xa<10 \lt x - a \lt 1 implies 0<(dc)(xa)<dc0 \lt (d - c)(x - a) \lt d - c and c<(dc)(xa)+c<dc \lt (d - c)(x - a) + c \lt d for all cc and dd in [x]\mathbb{Z}[x]. Let y=(dc)(xa)+cy = (d - c)(x - a) + c. Since there exists an element yy such that c<y<dc \lt y \lt d for all cc and dd, [x]\mathbb{Z}[x] is a dense linear order.

This means that the Dedekind completion of every Archimedean ordered integral domain extension of the integers is the integral domain of real numbers.

Examples

Archimedean ordered integral domains include

Non-Archimedean ordered integral domains include

Last revised on January 9, 2023 at 00:52:43. See the history of this page for a list of all contributions to it.